(*. (c) by Richard Lang, 2003 .*)

(* theory collecting all knowledge for LinearEquations

   created by: rlang 

         date: 02.10

   changed by: rlang

   last change by: rlang

             date: 02.10.20

*)

theory LinEq imports Poly Equation begin

axiomatization where

(*-- normalise --*)

  (*WN0509 compare PolyEq.all_left "[|Not(b=!=0)|] ==> (a = b) = (a - b = 0)"*)

  all_left:          "[|Not(b=!=0)|] ==> (a=b) = (a+(-1)*b=0)" and

  makex1_x:          "a^^^1  = a" and

  real_assoc_1:      "a+(b+c) = a+b+c" and

  real_assoc_2:      "a*(b*c) = a*b*c" and

(*-- solve --*)

  lin_isolate_add1:  "(a + b*bdv = 0) = (b*bdv = (-1)*a)" and

  lin_isolate_add2:  "(a +   bdv = 0) = (  bdv = (-1)*a)" and

  lin_isolate_div:   "[|Not(b=0)|] ==> (b*bdv = c) = (bdv = c / b)"

ML \<open>

val thy = @{theory};

val LinEq_prls = (*3.10.02:just the following order due to subterm evaluation*)

  Rule.append_rls "LinEq_prls" Rule.e_rls 

	     [Rule.Calc ("HOL.eq",eval_equal "#equal_"),

	      Rule.Calc ("Tools.matches", Tools.eval_matches ""),

	      Rule.Calc ("Tools.lhs"    , Tools.eval_lhs ""),

	      Rule.Calc ("Tools.rhs"    , Tools.eval_rhs ""),

	      Rule.Calc ("Poly.has'_degree'_in",eval_has_degree_in ""),

 	      Rule.Calc ("Poly.is'_polyrat'_in",eval_is_polyrat_in ""),

	      Rule.Calc ("Atools.occurs'_in",eval_occurs_in ""),    

	      Rule.Calc ("Atools.ident",eval_ident "#ident_"),

	      Rule.Thm ("not_true",TermC.num_str @{thm not_true}),

	      Rule.Thm ("not_false",TermC.num_str @{thm not_false}),

	      Rule.Thm ("and_true",TermC.num_str @{thm and_true}),

	      Rule.Thm ("and_false",TermC.num_str @{thm and_false}),

	      Rule.Thm ("or_true",TermC.num_str @{thm or_true}),

	      Rule.Thm ("or_false",TermC.num_str @{thm or_false})

              ];

(* ----- erls ----- *)

val LinEq_crls = 

   Rule.append_rls "LinEq_crls" poly_crls

   [Rule.Thm  ("real_assoc_1",TermC.num_str @{thm real_assoc_1})

    (*		

     Don't use

     Rule.Calc ("Rings.divide_class.divide", eval_cancel "#divide_e"),

     Rule.Calc ("Atools.pow" ,eval_binop "#power_"),

     *)

    ];

(* ----- crls ----- *)

val LinEq_erls = 

   Rule.append_rls "LinEq_erls" Poly_erls

   [Rule.Thm  ("real_assoc_1",TermC.num_str @{thm real_assoc_1})

    (*		

     Don't use

     Rule.Calc ("Rings.divide_class.divide", eval_cancel "#divide_e"),

     Rule.Calc ("Atools.pow" ,eval_binop "#power_"),

     *)

    ];

\<close>

setup \<open>KEStore_Elems.add_rlss 

  [("LinEq_erls", (Context.theory_name @{theory}, LinEq_erls))]\<close>

ML \<open>

val LinPoly_simplify = prep_rls'(

  Rule.Rls {id = "LinPoly_simplify", preconds = [], 

       rew_ord = ("termlessI",termlessI), 

       erls = LinEq_erls, 

       srls = Rule.Erls, 

       calc = [], errpatts = [],

       rules = [

		Rule.Thm  ("real_assoc_1",TermC.num_str @{thm real_assoc_1}),

		Rule.Calc ("Groups.plus_class.plus",eval_binop "#add_"),

		Rule.Calc ("Groups.minus_class.minus",eval_binop "#sub_"),

		Rule.Calc ("Groups.times_class.times",eval_binop "#mult_"),

		(*  Dont use  

		 Rule.Calc ("Rings.divide_class.divide", eval_cancel "#divide_e"),		

		 Rule.Calc ("NthRoot.sqrt",eval_sqrt "#sqrt_"),

		 *)

		Rule.Calc ("Atools.pow" ,eval_binop "#power_")

		],

       scr = Rule.EmptyScr});

\<close>

setup \<open>KEStore_Elems.add_rlss 

  [("LinPoly_simplify", (Context.theory_name @{theory}, LinPoly_simplify))]\<close>

ML \<open>

(*isolate the bound variable in an linear equation; 'bdv' is a meta-constant*)

val LinEq_simplify = prep_rls'(

Rule.Rls {id = "LinEq_simplify", preconds = [],

     rew_ord = ("e_rew_ord", Rule.e_rew_ord),

     erls = LinEq_erls,

     srls = Rule.Erls,

     calc = [], errpatts = [],

     rules = [

	      Rule.Thm("lin_isolate_add1",TermC.num_str @{thm lin_isolate_add1}), 

	      (* a+bx=0 -> bx=-a *)

	      Rule.Thm("lin_isolate_add2",TermC.num_str @{thm lin_isolate_add2}), 

	      (* a+ x=0 ->  x=-a *)

	      Rule.Thm("lin_isolate_div",TermC.num_str @{thm lin_isolate_div})    

	      (*   bx=c -> x=c/b *)  

	      ],

     scr = Rule.EmptyScr});

\<close>

setup \<open>KEStore_Elems.add_rlss 

  [("LinEq_simplify", (Context.theory_name @{theory}, LinEq_simplify))]\<close>

(*----------------------------- problem types --------------------------------*)

(* ---------linear----------- *)

setup \<open>KEStore_Elems.add_pbts

  [(Specify.prep_pbt thy "pbl_equ_univ_lin" [] Celem.e_pblID

      (["LINEAR", "univariate", "equation"],

        [("#Given" ,["equality e_e", "solveFor v_v"]),

          ("#Where" ,["HOL.False", (*WN0509 just detected: this pbl can never be used?!?*)

              "Not( (lhs e_e) is_polyrat_in v_v)",

              "Not( (rhs e_e) is_polyrat_in v_v)",

              "((lhs e_e) has_degree_in v_v)=1",

	            "((rhs e_e) has_degree_in v_v)=1"]),

          ("#Find"  ,["solutions v_v'i'"])],

        LinEq_prls, SOME "solve (e_e::bool, v_v)", [["LinEq", "solve_lineq_equation"]]))]\<close>

(*-------------- methods------------------------------------------------------*)

setup \<open>KEStore_Elems.add_mets

    [Specify.prep_met thy "met_eqlin" [] Celem.e_metID

      (["LinEq"], [],

        {rew_ord' = "tless_true",rls' = Atools_erls,calc = [], srls = Rule.e_rls, prls = Rule.e_rls,

          crls = LinEq_crls, errpats = [], nrls = norm_Poly},

        @{thm refl})]

\<close>

    (* ansprechen mit ["LinEq","solve_univar_equation"] *)

partial_function (tailrec) solve_linear_equation :: "bool \<Rightarrow> real \<Rightarrow> bool list"

  where

"solve_linear_equation e_e v_v =

(let e_e =((Try         (Rewrite     ''all_left''            False)) @@

           (Try (Repeat (Rewrite     ''makex1_x''            False))) @@

           (Try         (Rewrite_Set ''expand_binoms''       False)) @@

           (Try (Repeat (Rewrite_Set_Inst [(''bdv'', v_v)]

                                     ''make_ratpoly_in''     False))) @@

           (Try (Repeat (Rewrite_Set ''LinPoly_simplify''    False))))e_e;

     e_e = ((Try (Rewrite_Set_Inst [(''bdv'', v_v::real)]

                                     ''LinEq_simplify'' True)) @@

            (Repeat(Try (Rewrite_Set ''LinPoly_simplify''    False)))) e_e

 in Or_to_List e_e)"

setup \<open>KEStore_Elems.add_mets

    [Specify.prep_met thy "met_eq_lin" [] Celem.e_metID

      (["LinEq","solve_lineq_equation"],

        [("#Given", ["equality e_e", "solveFor v_v"]),

          ("#Where", ["Not ((lhs e_e) is_polyrat_in v_v)", "((lhs e_e)  has_degree_in v_v) = 1"]),

          ("#Find",  ["solutions v_v'i'"])],

        {rew_ord' = "termlessI", rls' = LinEq_erls, srls = Rule.e_rls, prls = LinEq_prls, calc = [],

          crls = LinEq_crls, errpats = [], nrls = norm_Poly},

        @{thm solve_linear_equation.simps})]

\<close>

ML \<open>Specify.get_met' @{theory} ["LinEq","solve_lineq_equation"];\<close>

end